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\title{Optimisation Project Abstract: Cloth Simulation}
\author{Roel Matthysen - s0202264 \\ Tuur Stuyck - s0200995 \\ 1e Master Wiskundige Ingenieurstechnieken}
\date{\today}

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\begin{document}

\maketitle

\section{Optimisation Objective}

In this project we wish to simulate the static behaviour of a piece of cloth. We model the piece of cloth as discrete points connected with springs, and we try to minimize the total potential energy, consisting of the energy stored in the springs, and the gravitational potential energy. In order to get realistic results, we use different interconnection schemes for the springs. We use structural springs (Figure \ref{fig:structural}), shear springs (Figure \ref{fig:shear}) and we may use bend springs (Figure \ref{fig:bend}) further on in the project. The structural springs ensure that the physical form of the cloth will be retained, the shear springs prevent each rectangle in the structural spring grid from collapsing into a diamond shape. The bend springs ensure that the cloth will display a realistic curvature instead of sharp angles. We will experiment with different springs i.e. both linear and non-linear springs. Combinations of these are also possible within the same piece of cloth. Another variant will be that no energy is stored in the spring when it is compressed, only when it is stretched. 

\begin{figure}[h]
\centering
  \subfloat[Structural]{\label{fig:structural}\includegraphics[width=0.3\textwidth]{img/structural.png}}                
  \subfloat[Shear]{\label{fig:shear}\includegraphics[width=0.3\textwidth]{img/shear.png}}
  \subfloat[Bend]{\label{fig:bend}\includegraphics[width=0.3\textwidth]{img/bend.png}}
  \caption{Different spring schemes}
\end{figure}

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\section{Decision variables}
The decision variables for this optimization problem are the three dimensional positions of the discrete points of the cloth.

\section{Constraints}
\subsection{Equality Constraints}
We will use equality constraints to keep the cloth fixed at certain points, e.g. in the corners. We can also keep a certain edge of the cloth fixed to model a curtain, or a flag. 
\subsection{Inequality Constraints}
We will use inequality constraints to model natural boundaries, like a ground constraint, or a fixed object. This will create a drape-like effect. These constraints will make the problem non-convex, so we will investigate the effects of different initial conditions.

\end{document}

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